Defining Simple Harmonic Motion (SHM)
Simple harmonic motion is a special kind of periodic motion in which the net restoring force is proportional to displacement from equilibrium and points toward equilibrium. In one dimension, F_net=-k_eff x, so a=-(k_eff/m)x=-ω²x. The word net matters: individual forces do not each have to be proportional to x as long as their sum produces the linear restoring relation about a stable equilibrium. Periodic motion by itself is not enough.
Topic sequence · 1 of 5
Learning objectives
Objective 1
Distinguish SHM from general periodic motion using the restoring-force criterion.
Objective 2
Identify equilibrium and define displacement from equilibrium correctly.
Objective 3
Connect F=-k_eff x to the differential equation d²x/dt²=-ω²x without jumping ahead to memorized sinusoidal formulas.
- Topic 2.5 Newton’s second law.
- Topic 2.8 spring forces.
- Topic 3.3 potential-energy equilibrium.
- Basic derivative meaning from Unit 1.
Definitions first
Representations you must read and create
Representation 1
Force-versus-displacement graph: a straight line through the origin with negative slope for ideal SHM.
Representation 2
Potential-energy curve near a stable minimum, where a quadratic approximation produces an approximately linear restoring force.
Representation 3
Motion diagram showing acceleration toward equilibrium on both sides.
Representation 4
Comparison table: periodic but non-SHM versus true SHM.
Core relationships and calculus
Variables: F N; k_eff N/m; x m
Use: defining one-dimensional linear restoring-force form
Variables: m kg
Use: Newton’s second law applied to SHM
Variables: x m; t s
Use: second-order differential equation for SHM
Variables: a m/s²; ω rad/s
Use: general SHM acceleration-displacement relation
Variables: ω rad/s
Use: spring-like oscillator with constant effective stiffness and inertia m
Newton’s second law gives m d²x/dt²=-k_eff x. Rearranged, d²x/dt²+ω²x=0 with ω²=k_eff/m. A sinusoidal function is a solution because differentiating it twice returns the same function with a factor -ω². Topic 7.3 develops the full position, velocity, and acceleration functions; Topic 7.1 focuses on recognizing the differential-equation signature.
The defining force law does not come from the fact that motion repeats. It comes from the local physics near stable equilibrium. If U(x) has a smooth minimum at x=0, then dU/dx=0 there; to lowest nonzero order U≈U0+½k_eff x², so F=-dU/dx≈-k_eff x. This explains why many small oscillations are approximately simple harmonic.
Topic 7.1 owns the defining net-force/acceleration condition. Full sinusoidal phase relations, extrema, resonance, and graph analysis belong to Topic 7.3; energy accounting belongs to 7.4; pendulum-specific derivations belong to 7.5. A small-angle pendulum may be cited as an SHM example, but its approximation must be stated rather than replaced by an arbitrary universal angle cutoff.
Physical examples
Example 1
An ideal horizontal mass-spring oscillator is SHM because F=-kx exactly.
Example 2
A small-angle pendulum is approximately SHM because sinθ≈θ makes the tangential restoring force proportional to displacement.
Example 3
Uniform circular motion is periodic but is not one-dimensional SHM of the orbiting object along its circular path.
Example 4
A nonlinear spring with F=-kx-cx³ is only approximately SHM for sufficiently small displacement if the cubic term is negligible.
Worked reasoning
Exam reasoning patterns
Reasoning pattern 1
Given a force-versus-position relation, use the net force—not one selected force—to decide whether the motion is SHM about the stated equilibrium.
Reasoning pattern 2
From a potential-energy curve, identify a stable minimum and show that the local force is approximately proportional to -x before labeling the motion SHM.
Reasoning pattern 3
Do not jump ahead to a sine/cosine formula until the physical restoring model has been established; Topic 7.3 owns phase and extrema analysis.
Reasoning pattern 4
For a vertical spring, measure x from the shifted equilibrium so the constant gravitational term cancels from the small-oscillation equation.
Deeper calculus and limiting-case check
The local quadratic expansion of a smooth potential near a stable minimum explains why SHM appears in many systems beyond literal springs. The effective stiffness is the curvature d²U/dx² at equilibrium.
Interactive SHM restoring-force check
For a linear restoring model F=-kx, compare displacement, restoring force, acceleration, and angular frequency.
Experimental connection
Attach a cart to springs and use a force sensor plus motion sensor. Measure net restoring force versus displacement about equilibrium. A straight-line fit with negative slope supports an SHM model; curvature or hysteresis indicates the linear restoring-force model is breaking down.
Connections across Physics C
Connection 1
Topic 3.3 connects stable potential-energy minima to restoring forces.
Connection 2
Topic 2.8 gives Hooke’s law as the canonical linear restoring force.
Connection 3
E&M Topic 13.6 will use the same SHM differential-equation structure for ideal LC circuits.
Related lessons and resources
Use these links when the current model depends on a prerequisite or becomes the foundation for the next model.
Common mistakes
Avoid this
Calling every repeating motion SHM. Periodicity is necessary but the linear restoring-force condition is the defining extra requirement.
Avoid this
Measuring x from an arbitrary origin instead of the equilibrium position.
Avoid this
Dropping the negative sign in F=-k_eff x; without it the force points away from equilibrium.
Avoid this
Assuming a pendulum is exact SHM at all amplitudes; the simple pendulum becomes SHM only in the small-angle approximation.
Avoid this
Using spring period formulas before identifying the effective restoring force and effective inertia.
Avoid this
Checking one force instead of the net force. SHM is diagnosed from the total force about equilibrium; constant forces may merely shift the equilibrium position.
Mastery check
- Define the main quantity without relying on an equation.
- Choose the correct representation and sign convention.
- Explain why the calculus relationship matches the physics.
- Design or interpret an experiment that could test the model.
Frequently asked questions
Is all periodic motion simple harmonic?
No. SHM additionally requires a net restoring force proportional to displacement and directed toward equilibrium.
Do all forces on the object have to look like -kx?
No. The criterion applies to the net force measured from equilibrium. A constant force, such as gravity on a vertical spring, can shift equilibrium without destroying the linear SHM model.
What is special about equilibrium?
The net force is zero there; for stable SHM a small displacement produces a net force back toward equilibrium.
Can SHM be approximate?
Yes. Many systems become approximately linear for sufficiently small displacement around a stable equilibrium; the approximation must be stated when the original restoring law is nonlinear.